Covering maps inject fundamental groups, with image detected by closed lifts of loops.Suggest a slogan
Let \(p\colon(\tilde X, \tilde x_0)\to(X, x_0)\) be a covering space. The induced map \(p_*\colon \pi_1(\tilde X, \tilde x_0) \to \pi_1(X, x_0)\) is injective, and its image consists of the classes of loops at \(x_0\) whose lifts to \(\tilde X\) starting at \(\tilde x_0\) are loops [@Hat02].